Recent advances

on interface constitutive laws

for Fracture Mechanics

Prof. Dr. Ing. Marco Paggi

Department of Structural, Geotechnical and Building Engineering

Politecnico di Torino, Torino, Italy

In collaboration with: Dr. Alberto Sapora

Supported by: Hannover, 20/08/2013

Acknowledgements

FIRB Future in Research 2010 Structural Mechanics Models for Renewable Energy Applications

ERC Starting Grant CA2PVM

Outlook & motivations

• Cohesive Zone Model for interface decohesion

• Extension to thermomechanical problems

1. Generalize thermoelastic contact analysis to fracture

(Zavarise et al. 1992; Wriggers and Zavarise, 1993)

2. Propose a simpler and physically consistent

formulation where the interface conductance depends on

crack opening, as an improvement and generalization of

other formulations

(Hattiangadi & Siegmund 2004; Yvonnet et al. 2010;

Özdemir et al., 2010)

Cohesive Zone Model (CZM)

s

gn

Double cantilever beam test

Applications

MgCa0.8 for biomedical stents

Cracking in

polycrystalline silicon

for PV modules

Paggi, Corrado, Composite Structures (2012)

Paggi, Sapora, Energy Procedia (2013)

Paggi, Lehmann, Weber, Carpinteri, Wriggers,

Schaper, Comp. Mat. Sci. (2013)

Hierarchical polycrystals Paggi, Wriggers, JMPS (2012)

Open issue:

Is the thermal CZM relating the heat flux to the temperature jump

independent of the mechanical CZM relating the cohesive

traction to crack opening?

CZM for stress and heat transfer

gn

s s

s=s(gn) Q=Q(DT)

Analogy between fracture & contact mechanics

pC

gn gnc

gn=0

pC

p

p

gn= gnc gn

No contact

Stress-free crack

Full contact

Perfect bonding

Partial decohesion

Partial contact

p=-s p=pC p=0

gn=0 gn

CONTACT

FRACTURE

Cohesive traction-separation relation

l0/R

Exponential decay inspired by

micromechanical contact models:

Greenwood & Williamson, Proc. R.

Soc. London (1966)

Lorenz & Persson, J. Phys. Cond.

Matter (2008)

gn/R

Heat flux-temperature gap relation

Q= -kint(gn) DT

Q

Qmax

gnc gn

Kapitza model

Interface conductance proportional to normal stiffness:

Barber, Proc. R. Soc. London (2003); Paggi & Barber, IJHMT (2011)

Xu & Needleman, Journal of the Mechanics and Physics of Solids (1994)

Özdemir et al. Computational Mechanics (2010)

Comparison with other CZM

Tractions Conductance

Formulation of the problem: 1

int

: ( )d ( )d ( )dS ( )dST T T

V V S S

δ V δ V δ δ = S w f w σ w σ w

T =S f 0

V: volume

S: surface

S: Cauchy stress tensor

f: body force vector

w: displacement vector

Strong form:

Weak form (PVW):

Formulation of the problem: 2

int

( )d ( ) d ( ) dS ( ) dST T

V V S S

δT V ρcT Q δT V δ q δT = - q q w

T Q ρcT- =q

V: volume

S: surface

q: heat flux vector

Q: heat generation

T: temperature

Strong form:

Weak form (variational form of the energy balance):

FE implementation: interface element

int

int dT

S

δG δ S= g p

int

int dT

S

δG δ S= g Cg

Gap vector:

g=(gt,gn,gT)T

Flux vector:

p=(τ,σ,q)T

Weak form for the interface elements:

Consistent linearization of the interface constitutive law

(quadratic convergence in the Newton-Raphson scheme):

=p Cg

Paggi, Wriggers, Comp. Mat. Sci. (2011); JMPS (2012)

FEA: constitutive matrix C

T

0

0

t n

t n

t n

τ τ

g g

σ σ

g g

q q q

g g g

=

C

If we consider kint=const (Kapitza’s model): 0t n

q q

g g

= =

Mechanical part

Thermoelastic part

with coupling

Tangent matrix for the Newton-Raphson algorithm:

Parametric analysis

TL<Ti

y*= y/L = 0.5

l0/R = 0.01

v = 0.1

Ti

- Implicit solution scheme in space and time

- Fully coupled thermomechanical problem (no staggered scheme)

- Use of an unsymmetric solver

Results 1

σ*max = 0.032, g*

nc=0.05

Temperature field vs. time (y*= y/L = 0.5)

Results 2

CZM vs. Kapitza model (kint=1/rint)

TEMPERATURE GAP DISPLACEMENT GAP

Neglecting thermoelastic coupling leads to very different

mechanical responses

T=70°C

d

T=40°C

micro-

crack

x

A preliminary application to photovoltaics

Staggered

solution

scheme

Conclusions

A thermo-mechanical CZM inspired by contact mechanics

between rough surfaces has been proposed:

> Interface conductivity dependent on crack opening

> Generalization of Kapitza model valid for perfectly bonded

interfaces

> Consistent thermo-mechanical formulation with only 4

parameters

> Novel results concerning the transient regime

Work in progress:

> Integration of the model in the multi-scale method (Paggi et

al., 2012) for the structural analysis of a PV module

Experimental laboratory

3D confocal-

interferometric

profilometer

(LEICA, DCM 3D)

SEM

(ZEISS, EVO MA15)

Testing stage

(DEBEN, 5000S)

Thermocamera

(FLIR, T640bx)

Photocamera for

EL tests

(PCO, 1300 Solar)

Testing machine &

thermostatic chamber

(Zwick/Roell, Z010TH)